The manufacturer of a fertilizer guarantees that, with the aid of the fertilizer, 65% of plantesl seeds will germinate. Suppose the manufacturer is correct. If ? seeds planted with the fertilizer are randomly selected, what is the probability that at least 6 of Item germinate?

Carry your intermediate computations to at least four decimal places, and round your answer to two decimal places.

To solve this problem, we'll use the binomial distribution, which models the number of successes (in this case, germinating seeds) in a fixed number of trials (seeds planted) with a constant probability of success on each trial.

Given:

- The probability of a seed germinating (\( p \)) = 0.65
- The number of trials (\( n \)) is not provided, so we will use \( n \) as a variable.
- We are asked to find the probability that at least 6 seeds germinate.

The probability that exactly \( k \) seeds germinate out of \( n \) seeds is given by the binomial probability formula:

\[
P(X = k) = \binom{n}{k} p^k (1-p)^{n-k}
\]

Where:
- \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \) is the binomial coefficient.
- \( p \) is the probability of success (0.65 in this case).
- \( 1-p \) is the probability of failure (0.35 in this case).
- \( X \) is the random variable representing the number of successes.

To find the probability that at least 6 seeds germinate, we need to sum the probabilities from 6 to \( n \):

\[
P(X \geq 6) = \sum_{k=6}^{n} P(X = k)
\]

However, since the value of \( n \) (the number of seeds) is not provided, we cannot calculate a numerical answer. If you provide the value of \( n \), I can compute the exact probability.

Would you like to provide the value of \( n \) or do you want to explore more details?

### Relative Questions:
1. What is the value of \( n \) in your problem?
2. Do you need help understanding the binomial distribution?
3. What is the significance of the 65% germination rate in other contexts?
4. How would the probability change if \( p \) were different?
5. How do you calculate factorials in the binomial coefficient?
6. Why is it necessary to compute intermediate values to four decimal places?
7. What does it mean to "round your answer to two decimal places"?
8. Can this problem be solved using a normal approximation?

**Tip:** When dealing with binomial probabilities, understanding the concept of "at least" can be made easier by considering the complementary probability (i.e., using \( P(X < 6) \)).

Learn how to calculate the probability of at least 6 seeds germinating using the binomial distribution. Understand the binomial probability formula and its application in predicting outcomes with a constant success probability. Detailed solution provided.

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